Braid group

review of symmetric groups

원소의 개수가 n인 집합의 전단사함수들의 모임
$n!$ 개의 원소가 존재함
대칭군의 부분군은 치환군(permutation group)이라 불림

presentation of symmetric groups

$S_n$
generators $\sigma_1, \ldots, \sigma_{n-1}$
relations
- ${\sigma_i}^2 = 1$
- $\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1$
- $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$

presentation of braid groups

$B_n$
generators $\sigma_1,...,\sigma_{n-1}$
relations (known as the braid or Artin relations):
- $\sigma_i\sigma_j =\sigma_j \sigma_i$ whenever $|i-j| \geq 2 $
- $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}$ for $i = 1,..., n-2$
Yang-Baxter equation (YBE)
For a solution of the YBE $\bar{R}$, we can construct a representation $\rho$ of the braid group by

$$ \rho : B_n \to \rm{Aut}(V^{\otimes n}) $$ where $\rho(\sigma_i)=\bar{R}_i$

There is also a natural surjective morphism from $B_n$ to the symmetric group $\mathfrak{S}_n$, given on the generators by $B_n\ni\sigma_i\mapsto s_i\in \mathfrak{S}_n$, $i=1,\dots,n-1$. For a braid $\beta\in B_n$, we denote $p_{\beta}$ its image in $\mathfrak{S}_n$, and refer to $p_{\beta}$ as to the underlying permutation of $\beta$.

examples

in a braid diagram, read from bottom to top and we number all strands of the braid with the indices it starts at the bottom.
read the braid word from left to right accordingly.
For instance, the braid word corresponding to the braid on the left of Figure ? is $\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_1^{-1}$

Markov moves

braid group version of Reidemeister moves

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxZ3NjMGpGUWI0QkE/edit

related items

encyclopedia

http://en.wikipedia.org/wiki/Braid_group

expositions

Abad, Camilo Arias. 2014. “Introduction to Representations of Braid Groups.” arXiv:1404.0724 [math], April. http://arxiv.org/abs/1404.0724.